If your opponent gets Spikes up, any sweeper can have their way with you. Without Spikes support, Moltres is no more broken than something like Arcanine or Houndoom, both of whom have Flash Fire and better tools for sweeping (Extremespeed and Nasty Plot respectively). If you can't remove the Spikes or keep the momentum of the match on your side, then you just need to revamp your strategy.
The extra power that Moltres has with 125 base special attack over Houndoom's 110 base and Arcanine's 110 base attack, coupled with a secondary STAB over Arcanine to hit certain "counters/checks" means that Moltres can power through its obstacles much more easily than both Arcanine and Houndoom. Looking at common counters/checks to Moltres, we have Chansey, Milotic, Regirock, and Altaria. With a layer of spikes and Stealth Rock up, these counters suddenly become very shaky. Looking at common counters/checks to Arcanine, we have Milotic, Blastoise, Regirock, Rhyperior, Altaria. Even with a layer of Spikes and Stealth Rock, Arcanine still has a bit of trouble getting through these pokemon, whereas Moltres can easily tear apart Milotic, Regirock, and Altaria with residual damage. Some calculations:
Naive "Morning Glory" Life Orb Arcanine Hidden Power Grass vs. standard bulky water Milotic: 21.4% - 25.4%
This will never reach a 2HKO with a layer of Spikes and Stealth Rock stripping off 25% on the switch. I'm using this set because very little Arcanine ran special attack EVs and most ran a Jolly or Adamant nature in January.
Same Arcanine Hidden Power Grass vs. standard Regirock: 26.9% - 31.9%
vs. 136 HP Rhyperior: 79.3% - 93.3%
One of the few cases where Spikes will help greatly.
Extremespeed vs. defensive Calm Altaria set: 25.4% - 29.9%
Can't 2HKO.
I guess Arcanine can effectively run sets like Specs and Band that allow it to power through its counters better, but the statistics show that hardly any Arcanine are running these sets.
+2 Timid Nasty Plot Life Orb Houndoom using Dark Pulse vs. bulky water Milotic: 63.4% - 74.8%
Houdoom just barely misses the KO with one layer of Spikes and Stealth Rock.
Same Houndoom vs. Regirock: 61% - 72%
Same Houndoom vs. defensive Calm Altaria set: 70.3% - 83.1%
Here's a nice calc for Houdoom. With Stealth Rock Altaria has a good chance of going down.
Plus Houndoom also has to deal with bullshit like Mach Punch Top and Hariyama, both of whom Moltres doesn't give a shit about. Houndoom must also deal with priority moves stopping its sweep, whereas Moltres can take Extremespeeds, Mach Punches, Vacuum Waves, and play around with Sucker Punchers a bit more easily than Houndoom.
Modest Life Orb Moltres using Air Slash vs. bulky water Milotic: 35.9% - 42.2%
Using Hidden Power Grass: 44.8% - 52.9%
The more common Modest set in January is able to easily take down Milotic with Air Slash and Hidden Power Grass.
Same Moltres using Hidden Power Grass vs. Regirock: 43.4% - 51.1%
A bit harder because Moltres likes to use a STAB move on switches. Fire Blast can deal 27.5% - 32.4% to tank Regirock.
Same Moltres using Air Slash vs. defensive Calm Altaria: 31.4% - 37.3%
Altaria has a good chance of surviving, but it can't do much back.
Basically, Arcanine and Houndoom are in no way comparable to Moltres in terms of wall breaking/sweeping power.